Question

example 9: sketch a picture of the rational function (f(x)) with the following properties: 1. (lim_{x
ightarrowinfty}f(x)=-infty) 2. (lim_{x
ightarrow-infty}f(x)=infty) 3. (lim_{x
ightarrow - 1^{-}}f(x)=-infty) 4. (lim_{x
ightarrow - 1^{+}}f(x)=-infty)

Explanation:

Step1: Analyze end - behavior

As $x\to\infty$, $\lim_{x\to\infty}f(x)=-\infty$ means the function goes down on the right - hand side. As $x\to-\infty$, $\lim_{x\to-\infty}f(x)=\infty$ means the function goes up on the left - hand side.

Step2: Analyze vertical asymptote behavior

Since $\lim_{x\to - 1^{-}}f(x)=-\infty$ and $\lim_{x\to - 1^{+}}f(x)=-\infty$, there is a vertical asymptote at $x = - 1$. The function approaches negative infinity from both the left and the right of $x=-1$.

Step3: Sketch the function

Draw a vertical asymptote at $x = - 1$. On the left - hand side of the vertical asymptote ($x\lt - 1$), the function starts from positive infinity as $x\to-\infty$ and goes down towards negative infinity as it approaches $x=-1$ from the left. On the right - hand side of the vertical asymptote ($x\gt - 1$), the function starts from negative infinity as it approaches $x = - 1$ from the right and continues to go down as $x\to\infty$.

Answer:

Sketch a rational function with a vertical asymptote at $x=-1$, going up as $x\to-\infty$, going down as $x\to\infty$, and going down as $x$ approaches $-1$ from both the left and the right.